Questions tagged [plane-geometry]

Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

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0answers
37 views

Thirteen-point conic and four-point line, are they new?

We know that Five points determine a conic and Two Points Determine a Line. Here I found a simple construct of a conic through $7$ points (in PS I note that how the conic through thirteen points) and ...
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1answer
48 views

What is the center of minimum distance of a region?

Suppose we have a compact plane region $R$ (not necessarily convex or connected). I am working in a problem which involves the point $p$ in $R$ that is, in average, the closest to every other point. ...
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176 views

Golden ratio in arbitrary triangle and how construct general problem

There are some new results above construction of the Golden ratio. Examples Odom's construction associated with an equilateral triangle, Dao's construction, Tran's construction associated with an ...
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85 views

A generic question on circles associated with a triangle

This question is inspired by two posed by P.Terzić (both given elegant synthetic proofs by F. Petrov). The starting point is a triangle $ABC$ and a triangle centre $G_1$. There are two classical ...
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1answer
50 views

Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim? Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...
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1answer
120 views

Does anyone know of an academic reference for this proof that the tangent to a hyperbola at a point bisects the angle between each focus to the point?

I am writing a report and as part of it I need to prove the property that for a point $P$ on a hyperbola, the tangent to the hyperbola at $P$ bisects the angle $\angle F_1PF_2$ where $F_1$ and $F_2$ ...
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2answers
148 views

A number characterizing the deviation of a triangle from the regular triangle

Given a triangle $\Delta$ with sides of length $a\le b\le c$, consider the number $$q=\frac{a^4+b^4+c^4}{(a^2+b^2+c^2)^2}$$ and observe that $\frac13\le q\le\frac12$ and the extremal values of $q$ ...
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1answer
128 views

Check if a polygon has an axis of symmetry in $O(n)$ time

Question: Is it possible to check if an $n$-gon has an axis of symmetry in $O(n)$ time? Note: An $O(n^2)$ algorithm is easy to see: it is easy to check if any given line is an axis of symmetry of the $...
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1answer
123 views

Collinearity in bicentric polygons

Can you provide a proofs for the following two claims? Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear. Claim ...
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1answer
128 views

Analytic or holomorphic extension of the ellipse perimeter function

Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$. Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^...
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133 views

Necessary and sufficient condition for tangential polygon to be cyclic

Can you prove or disprove the following claim? Claim. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided tangential polygon and let $B_1,B_2, \ldots ,B_n$ be the contact points of the ...
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1answer
149 views

Stability Question for Isotopies Between Compact Sets

Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$. Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a ...
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1answer
182 views

Point of concurrency [closed]

I am looking for the proof of the following claim: Claim: Let $\triangle ABC$ be an arbitrary triangle, $D$ its nine-point center and $E,F,G$ are the nine-point centers of the triangles $\triangle ...
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1answer
159 views

Proof of Denjoy-Riesz Theorem and Moore's Generalization?

The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane can be covered by an arc, i.e. an embedded image of $[0,1]$. Sometimes it's stated just for covering a Cantor Set,...
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75 views

Length of isoline $x(1-x)y(1-y)=c$

For the integral appearing in this answer, it may be beneficial to derive the length $L(c)$ of the isoline: $$x(1-x)y(1-y) = c,$$ where $x,y$ are ranging in $[0,1]$, and constant $c\in [0,\frac1{16}]$....
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1answer
78 views

On covering a disk by non-overlapping subdisks

I posted this question many years ago on math stackexchange but it did not get an answer. It had circulated as a puzzle in graduate school. A disk $D$ of radius $1$ contains disks $D_i$ ($i \ge 1$) of ...
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1answer
161 views

A generalization of Harcourt's theorem

This question is closely related to my previous question. Can you prove the claim given below? The following claim is a conjectured generalization of Harcourt's theorem. Claim. Let $A_1,A_2 \ldots ...
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Does this plane geometry theorem have a name (well-known)?

Consider three circles $(O_1)$, $(O_2)$, $(O_3)$. Denote the homothetic center of $\{$$(O_1)$, $(O_2)$$\}$ by $A$, the homothetic center of $\{$$(O_2)$, $(O_3)$$\}$ by $B$. Let $C$, $D$ be two points ...
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1answer
124 views

To cut a triangle into $n$ $p$-sided polygonal regions

Given any triangular region and two integers $n$ and $p$ which can be large and $p > 4$. It is needed to cut the triangle into $n$ $p$-gons (e.g., cut a triangle into 10 heptagons). Among the $p$-...
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1answer
206 views

A formula for the area of bicentric quadrilateral

Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals. Claim. Given bicentric quadrilateral $...
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1answer
119 views

The centroid, the first and second Napoleon points and $X(930)$ lie on a circle

Can you provide an elementary proof for the claim given below? Preliminary definitions: $X(110)=$ focus of Kiepert parabola. $X(137)=X(110)$ of orthic triangle . $X(930)=$ anticomplement of $X(137)$ . ...
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1answer
106 views

Four concyclic triangle centers

Can you prove the claim given below? Inspired by Lester's theorem I have formulated the following claim: Claim. Given any scalene triangle $\triangle ABC$ . Let $D$ be the reflection of incenter in ...
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2answers
389 views

On circles and ellipses drawn on an infinite planar square lattice

Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square ...
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1answer
57 views

Extending functional inequality from rectangles to parallelograms

Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ satisfying $f \geq 0$, $f(0,0) = 0$, $\frac{\partial{f}}{\partial{x}} \geq 0$, $\frac{\partial{f}}{\partial{y}} \...
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1answer
840 views

Happy ants never leave compact domain?

I am curious if the following seemingly simple question has an easy answer? Consider an ant population of $N$ ants that lives in $\mathbb R^2$. Each ant can be labeled by some coordinate $x\in \mathbb ...
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1answer
283 views

A generalization of Napoleon's theorem

Can you provide a proof for the following proposition? Proposition. Given an arbitrary $\triangle ABC$. The $\triangle AEB$, $\triangle BFC$ and $\triangle CDA$ are constructed on the sides of the $...
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229 views

Connecting a compact subset by a simple curve

Let $K$ be a compact subset of $\mathbb R^n$ with $n\ge 2$ (say if you like $n=2$, which is possibly sufficiently representative). Q: Does there exist a closed simple curve $u:\mathbb S^1\to\mathbb R^...
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1answer
90 views

Six concyclic points

Can you provide a proof for the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with excenters $J_A$,$J_B$ and $J_C$ . Let $G$ be the orthogonal projection of the $...
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2answers
144 views

Four concyclic points inside bicentric quadrilateral

Can you provide a proof for the following proposition: Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a ...
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572 views

Six points on an ellipse

Can you prove the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with centroid $G$. Let $D,E,F$ be the points on the sides $AC$,$AB$ and $BC$ respectively , such ...
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1answer
53 views

Vertices of 2 self-polar triangles lie on conic

I have conic $\gamma$ and two self-polar triangles $ABC$, $XYZ$ with respect to my conic. Why can I construct a one conic through $ABCXYZ$?
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166 views

Still very Bretschneider, isn't it?

Given a general convex quadrilateral with sides of lengths $a$, $b$, $c$, and $d$, the area is given by $$\begin{align*} K&=\frac{1}{4}\sqrt{4p^2q^2-(b^2+d^2-a^2-b^2)^2}\tag{1}\\&=\sqrt{(s-a)(...
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1answer
407 views

Weird geometry problem I found

I found this problem in a section of an old notebook, where I used to write down weird problems I came across and that I didn't know how to solve. Long story short, I rediscovered this notebook a week ...
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1answer
143 views

Constructing M-curves à la Hilbert

I have been reading some text about Harnack's theorem. The theorem basically says that for degree $d$, the maximal number of connected components in the real (projective) plane of a plane curve with ...
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4answers
2k views

Does this property characterize straight lines in the plane?

Take a plane curve $\gamma$ and a disc of fixed radius whose center moves along $\gamma$. Suppose that $\gamma$ always cuts the disk in two simply connected regions of equal area. Is it true that $\...
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2answers
140 views

What is the minimum number of triangle centers sufficient to unambiguously describe a triangle?

I am looking for a minimal number of properties describing a triangle so that these properties are invariant to the choice of a Cartesian coordinate system as well as to the order in which the ...
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Which subsets of the plane are similar to all their affine images?

A parabola P in the plane has the nice property that the image of P under any affine transformation is similar to P itself. Which other subsets of the plane have this property? I wondered aloud about ...
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1answer
726 views

Possible new theorem in plane geometry encompassing 5 famous geometry theorems

I am looking for a proof of a generalization Napoleon theorem, Bottema theorem and Brahmagupta theorem and van Aubel theorem, and Finsler–Hadwiger theorem in one configuration, as follows: Let four ...
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1answer
277 views

Discovered 240 new circles associated with Pascal's line

I am looking for a proof or a reference request for a problem as follows: Problem: Let a cyclic hexagon with sidelines $l_1$, $l_2$, $l_3$, $l_4$, $l_5$, $l_6$ and $l_1 \cap l_4 =A$, $l_3 \cap l_6 = ...
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1answer
102 views

On some centers of convex regions based on partitions

These questions are inspired by Yaglom and Boltyanskii's 'Convex figures'. Winternitz Theorem: If a 2D convex figure is divided into 2 parts by a line $l$ that passes through its center of gravity, ...
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1answer
62 views

Isotopy Classes of Non-Connected Planar Sets

I was just looking through some of my old questions on StackExchange and noticed that this one went totally unresolved. I still think it'd be useful as a lemma for plane topology if it were true, and ...
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2answers
354 views

Geometric construction of the fourth intersection points of two conics

In general, two conics in the plane intersect at most 4 points. Suppose three of those points are given as $A,B,C$. Then let $c_1$ be the conic passing through those three points and $D_1,E_1$. Let $...
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3answers
148 views

Can you measure the degree of uniformity of a 2d shape?

Is there a calculation that could take the points that make of the outline of a 2 dimensional shape and provide a numeric evaluation representative of the uniformity or symmetry of the shape. Such as ...
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25 views

Vertex configuration to tile repeat unit

I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...
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1answer
95 views

Identity map minus Cremona transformation

Let $ \delta $ be the triangle with vertices $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ in $\mathbb R^3$. It's a face of the standard octahedron. The Cremona transformation $$\mathcal C: (x, y, z) \mapsto - \...
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1answer
146 views

A claim on partitioning a convex planar region into congruent pieces

Let us define a perfect congruent partition of a planar region $R$ as a partition of it with no portion left over into some finite number n of pieces that are all mutually congruent (ie any piece can ...
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0answers
93 views

Perimeter points in triangle

Let $ABC$ denotes a triangle and $p(ABC)$ denotes its perimeter. We say two points $O_1$ and $O_2$ inside this triangle are perimeter points if there are points $a$, $b$ and $c$ on the sides $BC$, $AC$...
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2answers
206 views

Property of triangle centers

$M$ is the intersection of 3 cevians in the triangle $ABC$. $$AB_1 = x,\quad CA_1 = y,\quad BC_1= z.$$ It can be easily proven that for both Nagel and Gergonne points the following equation is true: $...
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62 views

Separating a certain planar region with an open set

I have a fairly specific question related to plane separation properties. I couldn't quite see how to use Phragmen–Brouwer properties to answer it because those kind of results generally apply to ...
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1answer
634 views

Brother of Japanese theorem for cyclic quadrilaterals

I am looking for a proof of a like result as follows and Higher-dimensional generalizations? Let $A, B, C, D$ be four point with lengths of $AB, BC, CD, DA$ are $a, b, c, d$ respectively. Let $F \in ...

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